I'm looking for a single word or short phrase for someone who intentionally burdens his or her self with a task or something that could be seen as a punishment purely to gain others' pity and make them feel as though they have wronged the person — sort of a "woe is me", martyr-like attitude. For instance, if a parent forces a child to do his or her chores against their will, the child might take an unusually long time to do it (making it seem like an overly arduous task) or might even intentionally break a piece of furniture in the process and claim "see what you made me did!" or something to that effect in order to seem like the victim, flipping the scenario's blame upon the parent.
Martyr it is (or play the martyr, if you want to be clearer):
1.1 A person who displays or exaggerates their discomfort or distress in order to obtain sympathy or admiration:
She wanted to play the martyr.
In the extreme, there is Munchausen syndrome.
Munchausen syndrome is a psychiatric factitious disorder wherein those affected feign disease, illness, or psychological trauma to draw attention, sympathy, or reassurance to themselves.
Munchausen syndrome is a health condition, and some argue that the person with the condition is not especially aware of the manipulation. For someone who is certainly aware of the deceit, you can use malingering.
Malingering is fabricating or exaggerating the symptoms of mental or physical disorders for a variety of "secondary gain" motives, which may include financial compensation (often tied to fraud); avoiding school, work or military service; obtaining drugs; getting lighter criminal sentences; or simply to attract attention or sympathy.
For less extreme situations, melodramatic is pejorative and clear.
emotional in a way that is very extreme or exaggerated
"Melodramatic." Merriam-Webster.com. Accessed April 4, 2016. http://www.merriam-webster.com/dictionary/melodramatic.
You might consider, emotional blackmail
Emotional blackmail and FOG, terms coined by psychotherapist Susan Forward, are about controlling people in relationships and the theory that fear, obligation and guilt ("FOG") are the transactional dynamics at play between the controller and the person being controlled. Understanding these dynamics are useful to anyone trying to extricate from the controlling behavior of another person, and deal with their own compulsions to do things that are uncomfortable, undesirable, burdensome, or self-sacrificing for others.